On 11/4/2010 4:10 PM, Marshall wrote:> On Nov 3, 11:40 pm, herbzet<herb...@gmail.com> wrote:>> Herman Jurjus wrote:>>> Marshall wrote:>>>> Herman Jurjus wrote:>>>>> herbzet wrote:>>>>>> Aatu Koskensilta wrote:>>>>>>> herbzet writes:>>>>>>>> Bill Taylor wrote:>>>>>>>>>>> Or whether the number 6 really exists. Does it?>>>>>>>>>> It *could* exist -- therefore, mathematically, it *does* exist.>>>>>>>>> This is a traditional and appealing idea. But just what is meant by>>>>>>> "could" here? What sort of possibility is involved?>>>>>>>> For rhetorical punch, I purposely left out the modifier, which is "logical".>>>>>>>> What logically could exist -- that is, what is not inherently self->>>>>> contradictory -- has mathematical existence.>>>>>>> Corollary: CH is false.>>>>> Proof: Since Cohen 1963 we know that it is logically consistent to>>>>> assume that there exists S, subset of P(N), equipollent neither to N nor>>>>> to P(N).>>>>>> Consistent with what? In what theory?>>>>> Co-consistent with ordinary mathematics, of course.>>> (I.e. with ZFC, and then also with any weaker theory.)>>>> Right -- the assumption here is that ordinary mathematics>> (i.e. ZFC, more or less) is itself consistent -- the ordinary>> and unremarkable gentleman's agreement.>> Ok. But again, all the "counterexamples" just amount to saying> that (most) theories have undecidable sentences, right?

It may be the case that the remark 'most theories have undecidable sentences' is used early on in the argument, but it's not the crux of what's being said.

Here's the same argument in terms of 'real' mathematics:

If you start a paper or discourse with"Let S be an uncountable subset of P(N), not equipollent to P(N)" (*)and you proceed from there, then, as long as you stick to ordinary mathematics, you'll never reach a contradiction (unless the ordinary mathematics that you use is in itself already inconsistent).

So, if mere absence of contradiction (using ordinary mathematical reasoning alone) is argument enough for mathematical existence, then you would be compelled to accept the mathematical existence of S, similar to that of the object '6'.

(*) And likewise, of course, with"Let f be a bijection from w_1 to P(N)"